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Cube
|bowers_acronym = cube |vertex_count = 8 |edges_length = 12l |surface_area = 6l^2 |surcell_volume = l^3 |vertices = 8 points |edges = 12 line segments |faces = 6 squares |cells = 1 cube |image1 = BowersCube.png|symmetry = Full octahedral symmetry (Oh)}} A cube (also known as a hexahedron) is the 3-dimensional hypercube. It is also the only platonic solid that can perfectly tessellate space by itself in a honeycomb, forming the cubic honeycomb. Under the elemental naming scheme it is called a geohedron. Among the Platonic solids the cube represents Earth since it is solid and unchanging. It has the Schläfli symbol \{4,3\} , meaning that it is made of squares, three of which meet at each vertex. It can also be represented by the Schläfli symbols { \{ \} }^{ 3 } as it is the product of three line segments, \{ 4\} \times \{ \} as it is the product of a square and a line segment (in other words, a square-based prism) and t\{ 2,4\} as it is a truncated square hosohedron. Its Bowers acronym is also "cube". Hypercube Product A cube can be expressed as the product of hypercubes in 3 different ways: \{4,3\} - cube As a regular cube, the subfacets and hypervolumes depend only on a single parameter, the edge length l. This is the most symmetrical form of the cube, and is a uniform, regular platonic solid. \{4\} \times \{\} - square prism As a square prism, created by the Cartesian product of a regular square and a line segment, the cube has square prismatic symmetry (D4h) with the abstract group Dih4 × Z2. The hypervolumes of a square prism depend on two parameters: the edge length a of the square, and the height of the prism b. The hypervolumes are: * edge length = 4 \left( 2a + b \right) * surface area = 2a \left( a + 2b \right) * surcell volume = a^2 b (when a=b, this becomes the regular cube.) \{\}^3 - line prism prism As a line prism prism, also known as a rectangular cuboid, created by the Cartesian product of three seperate line segments, the cube has digonal prismatic symmetry. The hypervolumes of a rectangular cuboid depend on three parameters, the lengths a, b and c of the line segments. The hypervolumes are: * edge length = 4 \left( a + b + c \right) * surface area = 2 \left( ab + ac + bc \right) * surcell volume = abc (when a=b, b=c xor a=c, this becomes the square prism. When a=b=c, this becomes the regular cube) Symbols A cube can be given various Dynkin symbols, including: *x4o3o (regular) *x x4o (square prism) *x x x(rectangular prism) *qo3oo3oq&#zx (2-coloring of vertices) *x2s4s, x2s8o (variations of the above) *s2s4x (two rectangles and 4 trapezoids) *xx4oo&#x (square frustum) *xx xx&#x (rectangle frustum) *oqoo3ooqo&#xt (triangular antitegum) *xxx oqo&#xt (prism of kite) *xx qo oq&#zx (rhombic prism) Structure and Sections The cube is composed of many squares stacked on each other, making it a prism with a square as the base. It is composed of two parallel squares linked by a ring of four squares. Three squares join at each corner. When viewed from a square face, it appears as a constant sized square. When viewed from an edge, it looks like a line expanding to a rectangle and back. Finally, when viewed from a corner, it is a point that expands into an equilateral triangle, then truncates to various hexagons, then goes back to a triangle (oriented the other way) which then shrinks. Hypervolumes * vertex count = 8 * edge length = 12l * surface area = 6l^2 * surcell volume = l^3 Subfacets * 8 points (0D) * 12 line segments (1D) * 6 squares (2D) * 1 cube (3D) Only 4D creatures and above could see all of a cube. Radii *Vertex radius: \frac{\sqrt{3}}{2}l *Edge radius: \frac{\sqrt{2}}{2}l *Face radius: 1/2l Angles *Dihedral angle: 90º Vertex coordinates The vertex coordinates of a cube of side length 2 are (±1,±1,±1). Equations All points on the surface of a cube of side 2 can be given by the equation \max(x^2,y^2,z^2) = 1 Notations *Toratopic notation: ||| *Tapertopic notation: 111 Related shapes *Dual: Octahedron *Vertex figure: Equilateral triangle, side length \sqrt{2} Coordinate System The coordinate system corresponding to the cube is called realm cartesian coordinates, with the three coordinates being \left(x, y, z\right) . The length elements of cartesian coordinates are simply dx , dy and dz , giving a line element of ds = dx \hat{x} + dy \hat{y} + dz \hat{z} with a length ds^2 = dx^2 + dy^2 + dz^2 . The surface elements, giving the changes in area for small changes in x, y and z, are dx dy , dx dz and dy dz . The volume element, giving the change in volume for small changes in x, y and z, is dx dy dz . See Also Category:Shape Category:3 dimensional Category:Hypercubes Category:Rotatopes Category:Polyhedra Category:Regular polyhedra Category:Uniform polyhedra Category:Toratopes Category:Tapertopes Category:Platonic solid